1. CSCI-455/552 Introduction to High Performance Computing Lecture 25

2. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 2 Solving a System of Linear Equations Objective is to find values for the unknowns, x0, x1, …, xn-1, given values for a0,0, a0,1, …, an-1,n-1, and b0, …, bn.

3. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 3 Solving a System of Linear Equations Dense matrices Gaussian Elimination - parallel time complexity O(n 2 ) Sparse matrices By iteration - depends upon iteration method and number of iterations but typically O(log n) Jacobi iteration Gauss-Seidel relaxation (not good for parallelization) Red-Black ordering Multigrid

4. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 4 Gaussian Elimination Convert general system of linear equations into triangular system of equations. Then be solved by Back Substitution. Uses characteristic of linear equations that any row can be replaced by that row added to another row multiplied by a constant. Starts at the first row and works toward the bottom row. At the ith row, each row j below the ith row is replaced by row j + (row i) (-aj,i/ ai,i). The constant used for row j is -aj,i/ai,i. Has the effect of making all the elements in the ith column below the ith row zero because

5. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 5 Gaussian elimination

6. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 6 Partial Pivoting If ai,i is zero or close to zero, we will not be able to compute the quantity -aj,i/ai,i. Procedure must be modified into so-called partial pivoting by swapping the ith row with the row below it that has the largest absolute element in the ith column of any of the rows below the ith row if there is one. (Reordering equations will not affect the system.) In the following, we will not consider partial pivoting.

7. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 7 Sequential Code Without partial pivoting: for (i = 0; i < n-1; i++) /* for each row, except last */ for (j = i+1; j < n; j++) { /*step thro subsequent rows */ m = a[j][i]/a[i][i]; /* Compute multiplier */ for (k = i; k < n; k++) /*last n-i-1 elements of row j*/ a[j][k] = a[j][k] - a[i][k] * m; b[j] = b[j] - b[i] * m; /* modify right side */ } The time complexity is O(n3).

8. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 8 Parallel Implementation

10. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 10 Pipeline implementation of Gaussian elimination

11. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 11 Strip Partitioning (p << n) Poor processor allocation! Processors do not participate in computation after their last row is processed.

12. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 12 Strip Partitioning(p << n)

13. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 13 Cyclic-Striped Partitioning (p << n) An alternative which equalizes the processor workload:

14. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 14 Cyclic-Striped Partitioning (p << n)

15. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 15 Grid Partitioning (p = n 2 )

16. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 16 Grid Partitioning (p = n 2 )

17. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 17 Block Grid Partitioning (p << n 2 )

18. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 18 Block Grid Partitioning (p << n 2 )

19. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 19 Block Grid Partitioning (p << n 2 )

20. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2nd ed., by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 20 Cyclic Grid Partitioning (p << n 2 )